Estimation of general, discrete-time FM processes

A general, frequency modulated (GFM) signal characterizes the vibrations produced by compressors, turbines, propellers, gears and other rotating machines in a dynamic environment. A GFM signal is defined as the composition of a real or complex, periodic or almost periodic function (the carrier) with a real, differentiable function (the modulation). This dissertation develops a frequency domain algorithm to estimate the parameters of one or more GFM signals in noise using the expectation-maximization (EM) algorithm. The primary advantages of this approach is that the ratios (harmonic numbers) of the carrier function do not need to be known a priori, multiple GFM signals can be estimated simultaneously and the algorithm exploits knowledge of the noise spectrum so that a separate normalization procedure is not required. ^ Chapter 2 of this dissertation introduces the EM method and derives a mixture PDF approximation to the periodogram. The Cramer-Rao lower bounds for unbiased estimates obtained from the periodogram of the parameters of one or more complex sinusoids are derived as well. In the last section of chapter 2, the mixture PDF approximation to the periodogram is extended to the estimation of the parameters of a stationary periodic signal, and this estimator is compared to five other methods from the literature. ^ Chapter 3 develops a mathematical representation for GFM signals and derives an approximate expression for the discrete Fourier transform (DFT) of a general, discrete-time FM signal. Chapter 4 derives the EM algorithm to estimate a single GFM signal in noise, and Chapter 5 extends the EM algorithm derived in chapter 4 to estimate the parameters of multiple GFM signals in noise.